On non-iterative training of a neural classifier part-I: Separation of points by planes

In this and in the next paper, an entirely novel method of supervised neural learning which is non-iterative is described. The process is as follows: Every data point which may be n-dimensional is first separated from every other point by hyperplanes, so that no two points are un-separated by at least one hyper plane. This is always possible in high dimension spaces because of the availability of many degrees of freedom. The separation is done by a newly discovered non-iterative algorithm and is described in this paper. Once the data points are all separated by planes, these planes are then used to classify the data points. If sufficient information is known as to which point belongs to which class then, one can regroup (or cluster) those points which belong to the same class. Since the equations to the planes which separate each point from each other, are now known, one can use some of these planes to separate each cluster (if needed one may have to add a few more planes). So that one has a set of planes which separate each cluster from one another. This paper describes a set of algorithms which perform this clustering and which discover the planes separating each cluster from another, these algorithms once again perform the task in a non-iterative manner. The planes that separate each cluster can then form the basis of a neural architecture. Thus in this manner one obtains a Neural architecture to solve the classification problem. This paper and the next describes the entire process. However, this first paper confines itself to the separation of points and describes the Separation Algorithm which can separate any number of points in n-dimensional space from one another by hyper planes. Given a set G of N_f points, along with their coordinates, in n-dimensional X-Space, the algorithm partitions all the N_f points by using planes such that no two points are left un-separated by some plane. And and the next paper deals with the clustering problem and the determination of a neural architecture. Example problems are solved which demonstrate the efficiency and practicability of this entirely new method of classification.